Aatish - Lesson 1 Quadratic and Simultaneous Equation

1. How to find roots of a quadratic

1.1

Three standard methods

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Three standard methods
  • Factorisation (when it factors nicely).
  • Completing the square (turns ax²+bx+c into a perfect square).
  • Quadratic formula x = (−b ± √(b²−4ac)) / (2a).
1.2

Worked examples for each method

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Worked examples for each method
  • Compare the steps side‑by‑side.
  • Always rewrite into ax² + bx + c = 0 first.

2. Solving a system that includes a quadratic

2.1

Simultaneous equations

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Simultaneous equations
  • Use substitution to reduce to a single quadratic.
  • Factorise to get possible values and check both.

3. Parabola essentials

3.1

What y = ax² + bx + c looks like

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What y = ax² + bx + c looks like
  • If a>0 the curve opens upwards; if a<0 it opens downwards.
  • The lowest/highest point is called the vertex (turning point).
3.2

When a > 0

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When a > 0
  • Parabola has a minimum at the vertex.
  • The axis of symmetry passes through the vertex.
3.3

When a < 0

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When a < 0
  • Parabola has a maximum at the vertex.
  • Same symmetry idea but curve opens down.
3.4

Vertex form when a > 0

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Vertex form when a > 0
  • Write as a(x+p)² + q; vertex is (−p, q).
  • q is the minimum value.
3.5

Vertex form when a < 0

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Vertex form when a < 0
  • Write as a(x+p)² + q; vertex is (−p, q).
  • q is the maximum value.

4. Graphing worked examples

4.1

Example 1: x‑intercepts and range

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Example 1: x‑intercepts and range
  • Find where y=0 to get intercepts.
  • Use symmetry to locate the vertex and state the range.
4.2

Example 2: rewrite to vertex form

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Example 2: rewrite to vertex form
  • Complete the square to read off the maximum point.
  • Then solve for intercepts if needed.

5. Absolute value of a quadratic

5.1

Sketch of y = |x² + 4x − 12|

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Sketch of y = |x² + 4x − 12|
  • Reflect any part below the x‑axis upwards.
  • Vertices occur where the original quadratic meets the x‑axis.
5.2

How many solutions for |x² + 4x − 12| = k

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How many solutions for |x² + 4x − 12| = k
  • Draw a horizontal line y=k and count intersections.
  • Special case at the peak/trough gives 3 solutions.

6. Discriminant and types of roots

6.1

Summary: b² − 4ac

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Summary: b² − 4ac
  • >0 two distinct real roots; =0 one repeated root; <0 no real roots.
  • Use this to predict how the graph meets the x‑axis.
6.2

How the graph meets the x‑axis

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How the graph meets the x‑axis
  • a>0 or a<0 cases shown for intersect/touch/miss.
  • Connect each picture to the discriminant condition.

7. Straight line and parabola

7.1

Intersect vs tangent vs no meet

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Intersect vs tangent vs no meet
  • Two points ⇒ b²−4ac>0; tangent ⇒ =0; no meet ⇒ <0.
7.2

Finding a tangent

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Finding a tangent
  • Set up the equations together and apply b²−4ac=0 to solve the parameter.

8. Parameter problems using the discriminant

8.1

No real roots: find the allowed q

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No real roots: find the allowed q
  • Rewrite to ax²+bx+c=0 and apply b²−4ac<0.
8.2

Real roots exist: find k

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Real roots exist: find k
  • Apply b²−4ac ≥ 0 and be careful when dividing by a negative number.
8.3

Equal roots: find h

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Equal roots: find h
  • Use b²−4ac = 0 to get a repeated root (touches x‑axis).
8.4

Touches x‑axis once: find p

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Touches x‑axis once: find p
  • Again b²−4ac = 0; expand and solve the resulting equation in p.
8.5

Does not meet x‑axis: range of k

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Does not meet x‑axis: range of k
  • For a>0, the whole graph is above x‑axis when b²−4ac < 0.
8.6

Intersects at two points: range of m

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Intersects at two points: range of m
  • Combine line and parabola into one quadratic in x; require b²−4ac>0.

9. Quadratic inequalities

9.1

< 0 means ‘below the x‑axis’

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< 0 means ‘below the x‑axis’
  • Solve the quadratic to get roots, then shade the region below the axis.
  • Answer lies strictly between the two roots.
9.2

≥ 0 means ‘on/above the x‑axis’

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≥ 0 means ‘on/above the x‑axis’
  • Outside the roots (or touching if discriminant=0).
  • Include the endpoints because of the ‘=’.